R15 - ENEE 241 02 READING ASSIGNMENT 15 Tue 04/07 Lecture...

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ENEE 241 02 * READING ASSIGNMENT 15 Tue 04/07 Lecture Topics: signal transformations and the DFT; duality; circular symmetry revisited Textbook References: sections 3.4, 3.5 Key Points: Time Domain Frequency Domain x = 1 N VX ←→ X = Wx P m x ←→ F - m X F m x ←→ P m X X ←→ N Rx Every DFT pair has a dual: if x ←→ X , then X ←→ N Rx . Circular conjugate symmetry in either domain (time or frequency) implies exclusively real values in the other domain, and vice versa. If a signal is both real-valued and circularly symmetric, then so is its spectrum. Theory and Examples: 1 . Let us take a closer look at circular shifts of the rows and/or columns of V and W . The diagonal matrix F (whose leading diagonal consists of the entries of v (1) ) is implicitly involved here. If 1 is the all-ones column vector, then we can write V = £ 1 F1 F 2 1 ... F N - 1 1 / A circular shift on the columns of V is obtained by right-multiplying it by P T = P - 1 , resulting in VP - 1 = £ F N - 1 1 1 F 1 1 ... F N - 2 1 / = £ F - 1 1 1 F 1 1 ... F N - 2 1 / = F - 1

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R15 - ENEE 241 02 READING ASSIGNMENT 15 Tue 04/07 Lecture...

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